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**Problem 8** Find \( z_x \) if \( z = \frac{1}{2x^2ay} + \frac{3x^{\frac{5}{2}}abc}{y} \). This problem requires the computation of the partial derivative of \( z \) with respect to \( x \). The expression for \( z \) involves two terms: a rational function and a term with a power of \( x \). **Problem 5**: Find \(\frac{dz}{dx}\) and \(\frac{dz}{dy}\) if \(z = (x^2 + x - y)^7\). This problem involves finding the partial derivatives of the function \(z = (x^2 + x - y)^7\) with respect to \(x\) and \(y\). ### Explanation: - **Partial Derivative \(\frac{dz}{dx}\)**: This involves differentiating the function with respect to \(x\), while keeping \(y\) constant. - **Partial Derivative \(\frac{dz}{dy}\)**: This involves differentiating the function with respect to \(y\), while keeping \(x\) constant. Understanding how to find these derivatives is crucial for applications in multivariable calculus, particularly when dealing with functions that depend on more than one variable.

**Problem 8** Find \( z_x \) if \( z = \frac{1}{2x^2ay} + \frac{3x^{\frac{5}{2}}abc}{y} \). This problem requires the computation of the partial derivative of \( z \) with respect to \( x \). The expression for \( z \) involves two terms: a rational function and a term with a power of \( x \). **Problem 5**: Find \(\frac{dz}{dx}\) and \(\frac{dz}{dy}\) if \(z = (x^2 + x - y)^7\). This problem involves finding the partial derivatives of the function \(z = (x^2 + x - y)^7\) with respect to \(x\) and \(y\). ### Explanation: - **Partial Derivative \(\frac{dz}{dx}\)**: This involves differentiating the function with respect to \(x\), while keeping \(y\) constant. - **Partial Derivative \(\frac{dz}{dy}\)**: This involves differentiating the function with respect to \(y\), while keeping \(x\) constant. Understanding how to find these derivatives is crucial for applications in multivariable calculus, particularly when dealing with functions that depend on more than one variable.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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find the partial derivatives. The variables
are restricted to a domain on which the function is defined.

**Problem 8** Find \( z_x \) if \( z = \frac{1}{2x^2ay} + \frac{3x^{\frac{5}{2}}abc}{y} \). This problem requires the computation of the partial derivative of \( z \) with respect to \( x \). The expression for \( z \) involves two terms: a rational function and a term with a power of \( x \).
Transcribed Image Text:**Problem 8** Find \( z_x \) if \( z = \frac{1}{2x^2ay} + \frac{3x^{\frac{5}{2}}abc}{y} \). This problem requires the computation of the partial derivative of \( z \) with respect to \( x \). The expression for \( z \) involves two terms: a rational function and a term with a power of \( x \).
**Problem 5**: Find \(\frac{dz}{dx}\) and \(\frac{dz}{dy}\) if \(z = (x^2 + x - y)^7\). This problem involves finding the partial derivatives of the function \(z = (x^2 + x - y)^7\) with respect to \(x\) and \(y\). ### Explanation: - **Partial Derivative \(\frac{dz}{dx}\)**: This involves differentiating the function with respect to \(x\), while keeping \(y\) constant. - **Partial Derivative \(\frac{dz}{dy}\)**: This involves differentiating the function with respect to \(y\), while keeping \(x\) constant. Understanding how to find these derivatives is crucial for applications in multivariable calculus, particularly when dealing with functions that depend on more than one variable.
Transcribed Image Text:**Problem 5**: Find \(\frac{dz}{dx}\) and \(\frac{dz}{dy}\) if \(z = (x^2 + x - y)^7\). This problem involves finding the partial derivatives of the function \(z = (x^2 + x - y)^7\) with respect to \(x\) and \(y\). ### Explanation: - **Partial Derivative \(\frac{dz}{dx}\)**: This involves differentiating the function with respect to \(x\), while keeping \(y\) constant. - **Partial Derivative \(\frac{dz}{dy}\)**: This involves differentiating the function with respect to \(y\), while keeping \(x\) constant. Understanding how to find these derivatives is crucial for applications in multivariable calculus, particularly when dealing with functions that depend on more than one variable.
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